All you can really do is contact the department to which you are applying and ask them if they have a procedure in place to appeal.
In my opinion, this is just another example of why the polygraph sucks and shouldn't be used in pre-employment screening.
A similar example that happens is when an applicant tells their BI they smoked pot twice in high school. On the polygraph the examiner tells the subject that they "showed deception" on the question about drug use (regardless of whether they actually did or not). The examiner, trying to be "helpful", asks if it is possible that the subject used pot three or four times instead of twice. Eager to avoid failing the polygraph, the subject agrees that maybe they did use it three times instead of two, even if they only remember two times.
Afterwards, the subject is mystified when they find out they failed the polygraph because, in the opinion of the examiner, they lied about prior drug use. The examiner sees it as: "They originally said they used pot twice, but then they admitted they used it more. That's a lie. The polygraph is great."
(Okay, I added in that last sentence by the polygraph examiner, because that's how I imagine they think. But the rest is pretty accurate.) ;)
Cogito ergo summopere periculosus.
Fiat justitia, ruat coelum.